How to select text between quotes, brackets... in Emacs?

Late to the party, but you can also use Evil mode, which does a bang-up job of Vim emulation, including the motion commands you mentioned.

On top of the toolkit

the following keys/commands are delivered:

(global-set-key [(super \))] 'ar-parentized-atpt)
(global-set-key [(super \])] 'ar-bracketed-atpt)
(global-set-key [(super \})] 'ar-braced-atpt)
(global-set-key [(super \")] 'ar-doublequoted-atpt)
(global-set-key [(super \')] 'ar-singlequoted-atpt)

That way with a couple of more chars known as delimiters will constitute commands.

ar-delimited-atpt will return the string around point found by nearest delimiter.

A group of more powerful commands allows re-using keys like that

(global-set-key [(control c)(\")] 'ar-doublequote-or-copy-atpt)
(global-set-key [(control c)(\')] 'ar-singlequote-or-copy-atpt)
(global-set-key [(control c)(<)] 'ar-lesser-angle-or-copy-atpt)
(global-set-key [(control c)(>)] 'ar-greater-angle-or-copy-atpt)

Here a doctring given as example:

ar-doublequote-or-copy-atpt is an interactive Lisp function in

It is bound to C-c ".

(ar-doublequote-or-copy-atpt &optional NO-DELIMITERS)

If region is highlighted, provide THING at point with doublequote(s),
  otherwise copy doublequote(ed) at point.
  With C-u, copy doublequote(ed) without delimiters. 
  With negative argument kill doublequote(ed) at point. 

From Emacs Documentation

Nearly all modes support “(,)” as parentheses, and most also support square brackets “[,]” and curly brackets “{,}”. However, you can make any pair of characters a parenthesis-pair, by using the following command:

(modify-syntax-entry ?^ "($")
(modify-syntax-entry ?$ ")^")

Also, take a look at this post How to mark the text between the parentheses in Emacs?. key combination given per this post

Try the key sequence C-M-u C-M-SPC (i.e., while holding the Control and Meta keys, press u and Space in sequence), which executes the commands backward-up-sexp and mark-sexp

The package expand-region is convenient for this. Calling er/expand-region with the point inside the quotes will mark the nearest word, and then calling it again will mark all the words inside the quotes. (Calling it a third time will expand the region to include the quotes.)

I have it bound to C-;.

(global-set-key (kbd "C-;") 'er/expand-region)

With this binding, pressng C-; C-; will highlight the text between the quotes.